Cryptology ePrint Archive: Report 2000/038

On the Complexity of Verifiable Secret Sharing and Multi-Party Computation

Ronald Cramer and Ivan Damgård and Stefan Dziembowski

Abstract:
We first study the problem of doing Verifiable Secret Sharing (VSS)
information theoretically secure for a general access structure. We
do it in the model where private channels between players and a
broadcast channel is given, and where an active, adaptive adversary
can corrupt any set of players not in the access structure. In
particular, we consider the complexity of protocols for this
problem, as a function of the access structure and the number of
players. For all access structures where VSS is possible at all, we
show that, up to a polynomial time black-box reduction, the
complexity of adaptively secure VSS is the same as that of ordinary
secret sharing (SS), where security is only required against a
passive, static adversary. Previously, such a connection was only
known for linear secret sharing and VSS schemes.

We then show an impossibility result indicating that a similar
equivalence does not hold for Multiparty Computation (MPC): we show
that even if protocols are given black-box access for free to an
idealized secret sharing scheme secure for the access structure in
question, it is not possible to handle all relevant access
structures efficiently, not even if the adversary is passive and
static. In other words, general MPC can only be black-box reduced
efficiently to secret sharing if extra properties of the secret
sharing scheme used (such as linearity) are assumed.